Test-time adaptation has emerged as a lightweight alternative to costly post-training for improving the reasoning capabilities of Large Language Models (LLMs) on downstream tasks. Predictive entropy provides a model-derived signal for such adaptation, guiding models toward higher-confidence reasoning states without external verifiers or reward models. However, higher confidence does not necessarily imply correctness, as LLMs may remain highly confident along incorrect reasoning trajectories. We observe that high-confidence reasoning is more likely to be correct when confidence remains stable under local perturbations. Based on this observation, we propose Test-Time Adaptation via Stability-Aware Confidence Optimization (TASCO), a framework that incorporates local stability into confidence-based test-time adaptation while keeping the LLM frozen. TASCO operationalizes local stability by optimizing a lightweight task-level prefix under two alternative perturbation strategies: Random Perturbation promotes distributional stability across trajectories induced by nearby perturbed prefixes, whereas Sharpness-Aware Perturbation targets worst-case local sensitivity. Experiments demonstrate that TASCO improves reasoning accuracy and token efficiency across diverse LLMs and reasoning benchmarks, while behavioral analyses show that it maintains stable confidence under local perturbations without prematurely concentrating the model's predictive distribution.

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